Problem: Determine how many solutions exist for the system of equations. ${-5x-y = -10}$ ${4x+y = -8}$
Answer: Convert both equations to slope-intercept form: ${-5x-y = -10}$ $-5x{+5x} - y = -10{+5x}$ $-y = -10+5x$ $y = 10-5x$ ${y = -5x+10}$ ${4x+y = -8}$ $4x{-4x} + y = -8{-4x}$ $y = -8-4x$ ${y = -4x-8}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -5x+10}$ ${y = -4x-8}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.